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In August 1908, a white mob attacked Springfield, Illinois’s Black community, destroying homes and businesses and lynching two Black men. National outrage over the violence helped spur the movement that created the NAACP the following year.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
 | This article may be too technical for most readers to understand. (August 2019) |
In applied mathematics, a grid or mesh is defined as the set of smaller shapes formed after discretisation of a geometric domain. Meshing has applications in the fields of geography, designing, computational fluid dynamics,[1] and more generally in partial differential equations numerical solving. The geometric domain can be in any dimension. The two-dimensional meshing includes simple polygon, polygon with holes, multiple domain and curved domain. In three dimensions there are three types of inputs. They are simple polyhedron, geometrical polyhedron and multiple polyhedrons. Before defining the mesh type it is necessary to understand elements (their shape and size).
An element of a mesh is one of the smaller shapes evoked in the introduction. The shape of the elements is of great importance in solving problems in computational fluid dynamics. They are typically based on aspect ratio i.e. the aspect ratio of element decide whether a particular element would be good to use or we should go for another element with different aspect ratio. For example, if the aspect ratio is large the speed of solver reduces while if this ratio is small the solver speed increases. Large aspect ratio has another limitation of leading to interpolation errors. But if the results vary with direction then we use large aspect ratio.
Most of the fluid flow equations are easily solved by discretizing procedures using the Cartesian coordinate system.[2] In this system the implementation of finite volume method is simpler and easier to understand. But most of the engineering problems deal with complex geometries that don’t work well in the Cartesian coordinate system. When the boundary region of the flow does not coincide with the coordinate lines of the structured grid then we can solve the problem by geometry approximation. Figures 1a. and 1b. shows how a cylinder can be approximated with the Cartesian coordinate system.
The curve geometry of cylinder in Cartesian coordinate system is approximated by using stepwise approximation. But this method requires large time and is very tedious to work with. Other than this problem there is one more problem which is the cells inside the solid part of the cylinder, which are called dead cells, are not involved in the calculations so they should be removed, otherwise they would consume extra space in computer or other resources. Stepwise approximation is not smooth and thus leads to significant error, though the grid can be refined by using a fine mesh to cover the wall region but this leads to waste of computer memory resources.
Therefore, there are limitation in using methods in computational fluid dynamics based on simple coordinate system (Cartesian or cylindrical) as these systems fails while modeling of complex geometries like that of an aerofoil, furnaces, gas turbine combustors, IC-engine etc.
a) Structured curvilinear grid arrangements (vertices having similar neighborhood).
b) Unstructured grid arrangements (vertices having variation in neighborhood).
Structured curvilinear grids
1) Grid points are identified at the intersection of coordinate line.
2) There are fixed number of neighboring grids for interior grid.
3) They can be arranged into an array and can be named by indices I, J, K f (In three dimensions).
These are also known as body-fitted grids and works on the principle of mapping the flow domain onto computational domain with simple shape. The mapping is quite tedious if it involves Complex geometry. In order to model this type of geometry we divide the flow region into various smaller sub domains. All these regions are meshed separately and joined up correctly with the neighbors. This type of arrangement is known as block-structured grid. This type of system is more flexible than the previous one. Two-dimensional structured mesh use quadrilaterals elements, while three-dimension meshes use hexahedra. There are two types of body-fitted coordinate grids:
a) Orthogonal curvilinear coordinate.
In orthogonal mesh the grid lines are perpendicular to intersection. This is shown in Figure 2.
b) Non–orthogonal coordinate.
Figure 3 shows non-orthogonal grids. The figure shows the grid lines do not intersect at 90-degree angle. In both these cases the domain boundaries coincide with the coordinate lines; therefore all the geometrical details can be incorporated. Grids can be refined easily to capture important flow features.
Comparison between Cartesian and curvilinear grids shows that in Cartesian grid cells are wasted in dealing with objects. The distribution of function is very fine in curvilinear grid. The resources required in curvilinear grids are less as compared to Cartesian grids thus saving much memory. Therefore, we can say that coarse grids are able to capture flow details efficiently.
Difficulties associated with the curvilinear grids are related to equations.[3]
While in Cartesian system the equation can be solved easily with less difficulty, but in curvilinear coordinate system it is difficult to solve the complex equations. Difference between various techniques lies in the fact that what type of grid arrangement is required and the dependent variable that is required in momentum equation. To generate meshes so that it includes all the geometrical features mapping is very important. In mapping physical geometry is mapped with computational geometry.
There are difficulties which we face in generating the body-fitted grids in geometries like IC engine combustion chamber. For example, the valve mapping in internal combustion engine is done very carefully so that the region of one type is mapped carefully with another type of region. There are regions where dense mesh is done deliberately to accommodate complex features. But this results in unnecessary grid resolution which leads to local variation of solution domain.
In this type of grid the domain is divided into different regions. Each region has different type of mesh structure. It is also possible that different coordinate system can be used for different regions. This makes the grids far more flexible. This also makes the refinement in the region where the geometry is to be captured more precise. Figure 4 shows the use of block grid technique. The advantages of this technique are that its generation is easy, the equations that we work with are easily discretized and curved boundaries are easily accommodated. Different blocks can be handled with required degree of fineness in mesh. GridPro can create beautiful multiblock structured meshes.
In far more complex geometries it is logical to use large number of blocks and therefore it leads to unstructured grids.[4] These are widely accepted in computational fluid dynamics because this gives us far more flexibility and computer resources are also efficiently utilized. In this case two-dimensional unstructured mesh uses triangle elements while three-dimensional uses tetrahedral elements. These are combination of small structured mesh arranged in unstructured pattern. In this type of grid each single cell is treated as a block. There is no structure of coordinate lines that is given by the grid. The advantage of this type of grids is that mesh can be refined wherever needed. This is based on the fact since the control volume can be of any shape therefore restriction on number of adjacent cell is lifted. This uses various combinations of cell shapes. Hybrid grid can also be used. Hybrid grids are those in which mixture of triangular and quadrilateral elements are used to construct grids. In three dimension combination of tetrahedral and hexahedral elements results in hybrid grid. Example of hybrid grid is shown in Figure 5. Unstructured grids reduces the time which is required for meshing and mapping. Therefore, grid generation is faster and easy. Various automatic techniques especially those associated with Finite Element Method also utilize unstructured grids. Mesh refinement and adaption is easy in unstructured grids.
This is done by two techniques namely smoothing and flipping. In mesh smoothing the location of mesh vertices are adjusted. While in Flipping the diagonals of triangulated quadrilateral are exchanged. Flipping improves the quality measure of triangles.
Source: Wikipedia. Article content is retrieved live through the MediaWiki API.
In applied mathematics, a grid or mesh is defined as the set of smaller shapes formed after discretisation of a geometric domain. Meshing has applications in the fields of geography, designing, computational fluid dynamics, and more generally in partial differential equations numerical solving. The geometric domain can be in any dimension. The two-dimensional meshing includes simple polygon, polygon with holes, multiple domain and curved domain. In three dimensions there are three types of inputs. They are simple polyhedron, geometrical polyhedron and multiple polyhedrons. Before defining the mesh type it is necessary to understand elements (their shape and size).
The SEC classification is the classification of consumers on the basis of parameters. Traditionally the two parameters used to categorize consumers were occupation and education of the chief wage earner (head) of the households. The SEC classification, created in 1988, was ratified by Market Research Society of India (MRSI), is used by most media researchers and brand managers to understand the Indian consuming class. Originally developed by IMRB International as a way of understanding market segments, and consumer behavior it was standardized and adopted by the Market Research Society of India in the mid-1980s as a measure of socio-economic class and is now commonly used as a market segmentation tool in India. In the older version, the SEC classification consists of two grids- The urban SEC grid, which uses education levels and occupational criteria of the chief wage earner (CWE) of a household as measures to determine socio-economic classification, and segments urban India into 7 groups (A1 to E2) and The rural SEC grid, which uses education and type of house (pucca, semi-pucca, and kaccha) as measures of socio-economic class, and segments rural India into 4 groups (R1, R2, R3, R4) This is based on the assumption that higher education leads to higher income thus higher consuming potential. But that this may not always be true. A trader or a retailer with no qualification can earn more income than a post-graduate executive, but SEC will categorize the traders/retailers not as SEC A1or A2. In order to combat this problem, the Government came up with the new SEC system on 3 May 2011. The new SEC system is based on two variables: Education of the chief earner. The options are illiterate, literate but no formal schooling or schooling unto 4 years, schooling between 5–9 years, high school pass, some college (including a diploma but not a graduate), graduate / post-graduate (general), graduate / post-graduate (professional) The number of consumer durables (pre-decided from a list of 11 items) owned by the family. The list of 11 items are: electricity connection, ceiling fan, LPG stove, two wheeler, color TV, refrigerator, washing machine, personal computer/laptop, car/jeep/van, air conditioner, agricultural land There are 12 grades in the new system starting from A1 to E3. These grids are used to determine the consumption preferences, and purchasing power of households, and are common tools used by social and business researchers working in India. The SEC grid does not use family income levels as a measure as this data is hard to collect and it has been demonstrated that education levels and occupation criteria in India are better determinants of consumer preference. The methodology used in these tools differs from the Household Potential Index, which measures consumption intensity.
Grid oscillations are oscillations in an electrical grid manifesting themselves in low-frequency (mostly below 1 Hz) periodic changes of power flow. These oscillations are a natural effect of negative feedback in power system control algorithms. During normal operation of the power grid, these oscillations, triggered by some change in the system, decay with time (are "damped"), and are mostly not noticeable. If the damping in the system is not sufficient, the amplitude of oscillations can grow, leading to a blackout. For example, shortly before the 1996 Western North America blackouts, the grid was oscillating with a frequency of 0.26 Hz for about 30 seconds after each disturbance. At some point a sequence of faults and opening of automatic protection relays caused loss of damping, eventually breaking the system into disconnected "islands" with many customers losing power. The other notable events involving oscillations were the Northeast blackout of 2003 and the 2009 sub-synchronous oscillations in Texas. While the theory and calculations tools for analyzing oscillations are available, pinpointing the source of instability in a real grid is frequently difficult as of the early 2020s. The oscillations are a normal occurrence, yet the difference in a flow as small as 10 MW is known to occasionally push the system from the stable mode with decaying oscillations into a situation where their amplitudes grow with time. The system operator frequently gets no warning that the grid is close to its damping limit.
The Köppen climate classification divides Earth's climates into five main climate groups, with each group being divided based on patterns of seasonal precipitation and temperature. The five main groups are A (tropical), B (arid), C (temperate), D (continental), and E (polar). Each group and subgroup is represented by a letter. All climates are assigned a main group (the first letter). All climates except for those in the E group are assigned a seasonal precipitation subgroup (the second letter). For example, Af indicates a tropical rainforest climate. The system assigns a temperature subgroup for all groups other than those in the A group, indicated by the third letter for climates in B, C, D, and the second letter for climates in E. Other examples include: Cfb indicating an oceanic climate with warm summers as indicated by the ending b, while Dwb indicates a semi-monsoonal continental climate, also with warm summers. Climates are classified based on specific criteria unique to each climate type. The Köppen climate classification is the most widely used climate classification scheme. It was first published by German-Russian climatologist Wladimir Köppen (1846–1940) in 1884, with several later modifications by Köppen, notably in 1918 and 1936. Later, German climatologist Rudolf Geiger (1894–1981) introduced some changes to the classification system in 1954 and 1961, which is thus sometimes called the Köppen–Geiger climate classification. As Köppen designed the system based on his experience as a botanist, his main climate groups represent a classification by vegetation type. In addition to identifying climates, the system can be used to analyze ecosystem conditions and identify the main types of vegetation within climates. Due to its association with the plant life of a given region, the system is useful in predicting future changes of plant life within that region. The Köppen climate classification system was modified further within the Trewartha climate classification system in 1966 (revised in 1980). The Trewartha system sought to create a more refined middle latitude climate zone, which was one of the criticisms of the Köppen system (the climate group C was too general).
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Shirley Chisholm, elected in 1968.